We study a certain class of weakly order preserving, non-injective (on an interval exactly ; called flat piece) circle maps which appear naturally in the study of Cherry flows on the two dimensions torus (see [1]}, [2]}, [3]}, [4]}), non-invertible continuous circle map (see [5]}) and of the dependence of the rotation interval on the parameter value for one-parameter families of continuous circle maps (see [6]}). The dynamics of circle maps with a flat interval has been intensively explored in the past years, see [7]}, [8]}, [5]}, [3]}, [11]}, [12]}.
